Finite Group Databasebeta

← #6434407

Subgroups — #6434407

Order 512.

1417 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34639C211yesmaximal, characteristic
2C2×#2647C211yesmaximal, characteristic, complemented
3#30395C214yesmaximal, complemented
4#30393C214yesmaximal, complemented
5#30391C214yesmaximal, complemented
6#30389C214yesmaximal, complemented
7#23745C211yesmaximal, characteristic, complemented
8#23734C211yesmaximal, characteristic
9#23670C212yesmaximal, complemented
10#23659C211yesmaximal, characteristic, complemented
11#23631C211yesmaximal, characteristic
12#23617C212yesmaximal
13#21159C212yesmaximal, complemented
14#21093C212yesmaximal
15#20429C211yesmaximal, characteristic, complemented
16C2×#428C2212yes—
17C2×#428C2212yes—
18C2×#423C2212yes—
19C2×#417C2212yes—
20C2×#417C2212yes—
21C2×C4×M16C2212yes—
22#2647—22no—
23#2647—22no—
24#2647C2214yes—
25#2647C2214yes—
26#2645C2212yes—
27#2645C2212yes—
28#2644C2212yes—
29#2644C2212yes—
30#2643—22no—
31#2643—22no—
32#2401—22no—
33#2401—22no—
34C2×#400—22no—
35C2×#398—22no—
36#2297C2211yescharacteristic
37#2297C2211yescharacteristic
38#2297C2211yescharacteristic
39#2296C2211yescharacteristic
40#2296C2211yescharacteristic
41#2296C2211yescharacteristic
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#376C2211yescharacteristic
47#2100C2214yes—
48#2099C2214yes—
49#2096C2212yes—
50#2095C2212yes—
51#2094C2212yes—
52#2093C2212yes—
53#2088C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2086C2214yes—
57#2085C2214yes—
58#2085C2214yes—
59#2074C2212yes—
60#2074C2212yes—
61#2073C2212yes—
62#2073C2212yes—
63#2068C2214yes—
64#2067C2214yes—
65#2066C2214yes—
66#2065C2214yes—
67#1944C2214yes—
68#1943C2214yes—
69#1942C2214yes—
70#1941C2214yes—
71#1936C2214yes—
72#1935C2214yes—
73#1934C2214yes—
74#1933C2214yes—
75#1768C2214yes—
76#1768C2214yes—
77#1736C2212yes—
78#1736C2212yes—
79#1736C2212yes—
80#1736C2212yes—
81#1728C2211yescharacteristic
82#1728C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C22×M16C2312yes—
85C22×M16C2312yes—
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#428—28no—
98#428—44no—
99C2×#107C2314yes—
100C2×#107C2314yes—
101C2×#106C2314yes—
102C2×#106C2314yes—
103#423—28no—
104C2×C4⋊C8C2314yes—
105C2×C4⋊C8C2314yes—
106#417—28no—
107#417—44no—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#102C2314yes—
111C2×#102C2314yes—
112C4×M16—44no—
113C2×#97C2314yes—
114C2×C4×C8C2314yes—
115#401—22no—
116#401—22no—
117#400—22no—
118#400—22no—
119#400—22no—
120#400—22no—
121#400—22no—
122#400—22no—
123#400—24no—
124#400—24no—
125#400—24no—
126#400—24no—
127#398—24no—
128#398—24no—
129#398—24no—
130#398—24no—
131#397—22no—
132#397—22no—
133#397—22no—
134#397—22no—
135#397—22no—
136#397—22no—
137#396—22no—
138#396—22no—
139#391C2312yes—
140#391C2311yescharacteristic
141#391C2311yescharacteristic
142#391C2312yes—
143#390C2312yes—
144#390C2311yescharacteristic
145#390C2311yescharacteristic
146#390C2312yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389C2314yes—
152#389C2314yes—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#389C2314yes—
160#389C2314yes—
161#386—21no—
162#386—22no—
163#386—22no—
164#386—21no—
165#385—21no—
166#385—22no—
167#385—22no—
168#385—21no—
169#384—22no—
170#384C2312yes—
171#384C2312yes—
172#382—22no—
173#382C2312yes—
174#382C2312yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C22⋊C4—22no—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183#376D412yes—
184#376D412yes—
185#376C2312yes—
186#376C2312yes—
187C2×#95—22no—
188C2×#95—22no—
189C2×#95—22no—
190C2×#95D412yes—
191C2×#95D412yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203#336C2312yes—
204C2×#41—24no—
205C2×#41—22no—
206C2×#41—24no—
207C2×#41—22no—
208C22×Q8C2411yescharacteristic
209C22×D4—21no—
210C22×D4C2411yescharacteristic
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2411yescharacteristic
219C2×M16—24no—
220C2×M16—24no—
221C2×M16—24no—
222C2×M16—24no—
223C2×M16—24no—
224C2×M16#4118yes—
225C22×C8C2414yes—
226C22×C8C2414yes—
227#117—24no—
228#117—22no—
229#117—24no—
230#117—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—84no—
250C2×C4⋊C4—84no—
251C2×C4⋊C4—84no—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2411yescharacteristic
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C2×C42C2412yes—
281C2×C42C2412yes—
282#107—28no—
283#107—44no—
284#106—28no—
285#106—44no—
286C4⋊C8—28no—
287C4⋊C8—28no—
288#103—28no—
289#103—44no—
290#102—28no—
291#102—44no—
292#97—44no—
293C4×C8—44no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95C2×D412yes—
313#95C2×D412yes—
314#95C2×D412yes—
315#95C2×D412yes—
316#95C2×D412yes—
317#95C2×D412yes—
318#95C2×D412yes—
319#95C2×D412yes—
320C24—21no—
321C24C22×D411yescharacteristic
322#41—48no—
323#41—48no—
324C2×Q8—22no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—21no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4114yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C42−1+414yes—
371C22×C42−1+412yes—
372C22×C4C2511yescharacteristic
373M16—216no—
374M16—48no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8C2×#4118yes—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C42—24no—
414C42—24no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—42no—
424C42—42no—
425C42—42no—
426C23—22no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23#53212yes—
434C23#53612yes—
435C23#53712yes—
436C23C23×D411yescharacteristic
437Q8—42no—
438Q8—42no—
439D4—42no—
440D4—42no—
441D4—84no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—88no—
484C2×C4—88no—
485C2×C4—84no—
486C2×C4#53212yes—
487C2×C4#53612yes—
488C2×C4#53712yes—
489C2×C4#53918yes—
490C2×C4#54118yes—
491C2×C4C22×#4112yes—
492C2×C4C2×H(5,𝔽2)11yescharacteristic
493C2×C4C2×2−1+412yes—
494C2×C4C2×2−1+411yescharacteristic
495C8—28no—
496C8—44no—
497C22—22no—
498C22—22no—
499C22—24no—
500C22—24no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—82no—
505C22—82no—
506C22#266414yes—
507C22C2×#53211yescharacteristic
508C22C2×#53611yescharacteristic
509C22C2×#53711yescharacteristic
510C22#358512yes—
511C22#358612yes—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—81no—
519C4—81no—
520C4—84no—
521C4—81no—
522C4—82no—
523C4—82no—
524C4—84no—
525C4#266414yes—
526C4C2×#53912yes—
527C4C2×#54112yes—
528C2—24no—
529C2—161no—
530C2C2×#266411yescharacteristic
531C2#6070411yescharacteristic
532C2#6070511yescharacteristic
533C2#6086712yes—
534C2#6091912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2647C2semidirect
3. #30395C2semidirect
4. #30393C2semidirect
5. #30391C2semidirect
6. #30389C2semidirect
7. #23745C2semidirect
9. #23670C2semidirect
10. #23659C2semidirect
13. #21159C2semidirect
15. #20429C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#346392
C2×#26472
#303952
#303932
#303912
#303892
#237452
#237342
#236702
#236592
#236312
#236172
#211592
#210932
#204292